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RoseWind [281]
3 years ago
11

Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of

F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = xy i + yz j + zx k S is the part of the paraboloid z = 8 − x2 − y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation
Mathematics
1 answer:
Maru [420]3 years ago
6 0

Parameterize S{/tex] by[tex]\vec s(u,v)=u\,\vec\imath+v\,\vec\jmath+(8-u^2-v^2)\,\vec k

with 0\le u\le1 and 0\le v\le1.

Take the normal vector to S to be

\vec s_u\times\vec s_v=2u\,\vec\imath+2v\,\vec\jmath+\vec k

Then the flux of \vec F across S is

\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_0^1\int_0^1\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^1\int_0^1(uv\,\vec\imath+v(8-u^2-v^2)\,\vec\jmath+u(8-u^2-v^2)\,\vec k)\cdot(2u\,\vec\imath+2v\,\vec\jmath+\vec k)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^1\int_0^1\bigg(2u^2v+(u+2v^2)(8-u^2-v^2)\bigg)\,\mathrm du\,\mathrm dv=\boxed{\frac{1553}{180}}

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I don’t understand lol
pogonyaev

Answer:

3*r = 3

Step-by-step explanation:

\dfrac{24}{r}=3\\\\24 = r*3\\\\\dfrac{24}{3}=r\\\\6 = r6 is a solution.

3*r=3\\\r =\dfrac{3}{3}\\\\r = 1\\

6 is not a solution.

7r = 42\\\\r =\dfrac{42}{7}\\\\r = 6

6 is a solution

8 0
2 years ago
Read 2 more answers
Thank you very much anh you
Neko [114]

Hello from MrBillDoesMath!

Answer:

See discussion below

Discussion:

#4

The sum of the interior angles of a polygon of n sides is (n-2)* 180. As the polygon in #4 has 7 sides , the sum of the interior angles is (7-2)*180 = 5 * 180 = 900.

We can also compute the sum of the interior angles by adding up each of the angles (expressed in terms of x)  shown in #4. Starting at the top of the polygon and proceeding counterclockwise gives:

900 =  (8x +34) + (10x+21)+(9x+30)+(7x+45)+(5x+44)+(12x+13)+(6x+29)  =>

           (8x + 10x+9x+7x+5x+12x+6x) + (34+21+30+45+44+13+29)  =

           57x                                            + 216

This simplifies to

900 =  57x + 216                => subtract 216 from both sides

900-216 = 57x                    => 900-216= 684

684= 57x                            => divide both sides by 57

684/57  = x                         => as 684/57 = 12

x = 12

The diagram has makes repeated use of the same variable for vertices. That is, the diagram shows 4 P's, 1 Q, 2 R's, 0 S's, and 1 T making it impossible to determine the values of some angles.  For example, consider m RST -- whatever that means.

#5 uses the same ideas but the vertices are properly  labeled (I'll leave the grunt of determining the  individual angles to you but here are the main points).

Sum of interior angles = (7 -2) * 180 = 5 * 180 = 900.

900 =  (8x+62) + (7x+66)+(5x+83)+(6x+55)+(10x+42)+(8x+45)+(4x+67)  =>

900 =   (8x + 7x +5x+6x+10x+8x+4x) + (62+66+83+55+42+45+67)

900=     48x                                        +  420             =>

900 -420 = 48x                                                          =>   subtract 420

480  = 48x                                                                  =>   divide by 48

x = 480/48 = 10

Thank you,

MrB

6 0
3 years ago
My sis need help with solutions
Sladkaya [172]

Answer:

Option B (2, -2) is correct.

Step-by-step explanation:

Have a great day! :)

and can I plz have brainliest?

7 0
3 years ago
Let a1equals[Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column 2 3rd Row 1st Column negative 1 EndMatrix ]​, a2equals[
Zolol [24]

Answer:

For h= 25,  b in the plane spanned by a1 and a2​

Step-by-step explanation:

a1= \left[\begin{array}{c}1\\2\\-1\end{array}\right] \\a2 = \left[\begin{array}{c}-7\\-7\\2\end{array}\right] \\\\b   = \left[\begin{array}{c}3\\-22\\h\end{array}\right]

we have to find value of h for which  b in the plane spanned by a1 and a2.

For this the linear systems given by the  following augmented matrix must be consistent.

\left[\begin{array}{cc|c}1&-7&3\\2&-7&-22\\-1&2&h\end{array}\right]

Reduce the augmented matrix into row echelon form:

R_{2} - 2R_{1} , R_{3} + R_{1}\\\\\left[\begin{array}{cc|c}1&-7&3\\0&7&-28\\-0&-5&h+3\end{array}\right]\\\\7R_{3}+5R_{2}\\\\\left[\begin{array}{cc|c}1&-7&3\\0&7&-28\\-0&0&7h-175\end{array}\right]

For system to be consistent:

                         7h-175 =0\\7h=175\\h=25

6 0
3 years ago
4
Bad White [126]

Answer:

D

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
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